Comparison of canonical periods under base change

In this paper we prove the canonical period of a Hilbert modular form with respect to the base change of a real quadratic extension differs from the square of its own canonical period only by a $p$-adic unit under some conditions. We prove this by proving a specific version of anticyclotomic Iwasawa main conjecture for Hilbert modular forms.

💡 Research Summary

The paper studies the behavior of the canonical period attached to a Hilbert modular newform under base change to a totally real quadratic extension. Let F be a totally real field and f a Hilbert newform of parallel even weight k with trivial central character. For a totally real quadratic extension F′/F, denote by f′ the base‑change of f to F′. The authors conjecture that the canonical period Ω_can(f′) equals the square of Ω_can(f) up to a p‑adic unit, i.e. Ω_can(f′)=Ω_can(f)^2·u with u∈O_f^×. The main theorem proves this conjecture under a collection of technical hypotheses.

Key hypotheses

• The prime p is unramified in both F and F′, satisfies p≥max(k+2,7), and is coprime to the level n_f.
  - f is ordinary at all primes above p.
• When